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User blog:Wythagoras/Dollar function: final version
Inspired by Hyp cos I decided to remake everything exepted for Bracket Notation. Which rule should you use? #If there is nothing after the $, use rule 1 #If there are any non-nested non-subscript numbers, use rule 2 #If there are any non-nested non-subscript 0's, use rule 3 #If there are any non-nested non-subscript b's, use rule 4 #If the previous things doesn't apply but the lowest level bracket can be solved with normal bracket notation: ##Search in the bracket for the least nested lowest level bracket or number ###If it is a 0: ####If the zero is the only content, use rule 3 ####Otherwise, use rule 5 ###If it is another number, use rule 4 ###If it is a bracket: Return to step 5 #If the lowest level bracket can be solved with extended bracket notation: ##Is the number in the typed bracket a 0, use rule 6 and the subrule if needed ##Otherwise, use rule 4 Extended Bracket Notation This works now like the Buchholz hydra, and the limit is \(\psi(\psi_I(0))\) 1. If there is nothing after the $, the array is solved. The value of the array is the number before the $. 2. \(a\$b\bullet=(a+b)\$\bullet\) 3. \(a\$\circ0\bullet\circ=a\$\circ a\bullet\circ\) 4. \(a\$\circ\bullet+1_c\bullet\circ=a\$\circ\bullet_c\bullet_c...\bullet_c\bullet_c\bullet\circ\) with a \(\bullet\)'s 5. If the bracket contains a zero and the bracket has other content, you can remove the zero. 6. If the active bracket has level k, search for the least nested bracket with level (k-1) with the same array in it. S1: The outermost bracket is always level 1 S2: If there is no bracket with level (k-1), add it. Linear Array Notation Here are no $, but this are just rules what you should do if that kind of array is the lowest level array. 7. \(b\bullet,c = [0,c-1_{b-1\bullet,c1}]\) 8. To diagonalize in the nth position with bracket types, you must use \(\underbrace{0,0...0,1}_n_k\) They diagonalize in the last entry. 9. \(\diamond,b\bullet,c,\bullet = [[\diamond,\diamond,b\bullet,c-1,\bullet_{\diamond,b-1\bullet,c,\bullet},c-1,\bullet]\) 10. \(0,c,bullet = 0\) S3. Zeroes at the and of the array must be removed Analysis \([[0,1]]\) has level \(\psi(\psi_I(0))\) \([[00,1]]\) has level \(\psi(\psi_I(1))\) \([[1,1]]\) has level \(\psi(\psi_I(\omega))\) \([[[0_2],1]]\) has level \(\psi(\psi_I(\varepsilon_0))\) \([[[[0_2]_2],1]]\) has level \(\psi(\psi_I(\zeta_0))\) \([[[[0_3]_2],1]]\) has level \(\psi(\psi_I(\varphi(\omega,0)))\) \([[[[0,1]],1]]\) has level \(\psi(\psi_I(\psi(\psi_I(0))))\) \([[[[[[0,1]],1]],1]]\) has level \(\psi(\psi_I(\psi(\psi_I(\psi(\psi_I(0))))))\) \([[0_2,1]]\) has level \(\psi(\psi_I(\Omega))\) \([[0_{0},1]]\) has level \(\psi(\psi_I(\Omega_\omega))\) \([[[0,1],1]]\) has level \(\psi(\psi_I(\psi_I(0)))\) \([[0,1_2,1]]\) has level \(\psi(\psi_I(I))\) \([[1,1_2,1]]\) has level \(\psi(\psi_I(I\omega))\) \([[[0_2,1]_2,1]]\) has level \(\psi(\psi_I(I\Omega))\) \([[[0,1,1]_2,1]]\) has level \(\psi(\psi_I(I\psi_I(0)))\) \([[[0,1_2,1]_2,1]]\) has level \(\psi(\psi_I(I^2))\) \([[[1,1_2,1]_2,1]]\) has level \(\psi(\psi_I(I^\omega))\) \([[[[0,1_2,1]_2,1]_2,1]]\) has level \(\psi(\psi_I(I^I))\) \([[0,1_3,1]]\) has level \(\psi(\psi_I(\varepsilon_{I+1}))\) \([[0,1_4,1]]\) has level \(\psi(\psi_I(\varphi(\omega,I+1)))\) \([[[0,1_40,1_4]_3,1]]\) has level \(\psi(\psi_I(\Omega_{I+1}))\) \([[[0,1_50,1_5]_4,1]]\) has level \(\psi(\psi_I(\Omega_{I+2}))\) \([[0,2]]\) has level \(\psi(\psi_{I_2}(0))\) \([[0,0]]\) has level \(\psi(\psi_{I_\omega}(0))\) \([[0,0_2]]\) has level \(\psi(\psi_{I_\Omega}(0))\) \([[0,0,1]]\) has level \(\psi(\psi_{I_{\psi_I(0)}(0))\) \([[0,0,2]]\) has level \(\psi(\psi_{I_{\psi_{I_2}(0)}(0))\) \([[0,0,1_2]]\) has level \(\psi(\psi_{I_{I}(0))\) \([[0,0,1]]\) has level \(\psi(\psi_{\chi(1)}(0))\) \([[0,00,1]]\) has level \(\psi(\psi_{\chi(1)}(1))\) \([[0,0_2,1]]\) has level \(\psi(\psi_{\chi(1)}(\Omega))\) \([[0,0,0,1_2,1]]\) has level \(\psi(\psi_{\chi(1)}(\chi(1)))\) \(0,0,2\) has level \(\psi(\psi_{\chi(2)}(0))\) \(0,0,3\) has level \(\psi(\psi_{\chi(3)}(0))\) \(0,0,[0]\) has level \(\psi(\psi_{\chi(\omega)}(0))\) \([[0,0,0_2]]\) has level \(\psi(\psi_{\chi(\Omega)}(0))\) \(0,0,[0,1]\) has level \(\psi(\psi_{\chi(\psi_I(0))}(0))\) \([[0,0,0,0,1_2]]\) has level \(\psi(\psi_{\chi(M)}(0))\) \([[0,0,0,1]]\) has level \(\psi(\Psi_{\Xi(3,0)}(0))\) \([[0,0,0,0,1]]\) has level \(\psi(\Psi_{\Xi(4,0)}(0))\) limit of linear arrays has level \(\psi(\Psi_{\Xi(\omega,0)}(0))\) Category:Blog posts